Content uploaded by Peter Bjorklund
Author content
All content in this area was uploaded by Peter Bjorklund on Aug 15, 2014
Content may be subject to copyright.
Resource Optimization of Spatial TDMA in Ad Hoc Radio
Networks: A Column Generation Approach
Patrik Bj
¨
orklund, Peter V
¨
arbrand and Di Yuan
Department of Science and Technology, Link
¨
oping University
SE-601 74 Norrk
¨
oping, Sweden
patbj, petva, diyua@itn.liu.se
Abstract—Wireless communications using ad hoc networks are
receiving an increasing interest. The most attractive feature of ad
hoc networks is the flexibility. The network is set up by a number
of units in an ad hoc manner, without the need of any fixed infras-
tructure. Communication links are established between two units
if the signal strength is sufficiently high. As not all pairs of nodes
can establish direct links, the traffic between two units may have
to be relayed through other units. This is known as the multi-hop
functionality.
Design of ad hoc networks is a challenging task. In this paper
we study the problem of resource allocation with spatial TDMA
(STDMA) as the access control scheme. Previous work for this
problem has mainly focused on heuristics, whose performance is
difficult to analyze when optimal solutions are not known. We de-
velop, for both node-oriented and link-oriented allocation strate-
gies, mathematical programming formulations for resource op-
timization. We further present a column generation approach,
which, in our numerical experiments, constantly yields optimal
or near-optimal solutions. Our results provide important bench-
marks when evaluating heuristic on-line algorithms for resource
optimization using STDMA.
Index Terms – Ad hoc networks, STDMA, node and link assign-
ment, column generation
I. INTRODUCTION
An ad hoc network is characterized by a collection of
radio units with a wireless interface forming temporary
connections. No fixed infrastructure is involved in the
communication. Instead, two radio units can establish a
direct communication link, if the signal-to-noise ratio is
high enough. Two radio units far away from each other
may communicate, if the units between them are partici-
pating in the ad hoc network, and are willing to forward
packets for them (so called the
multi-hop functionality
).
Multi-hop radio networks have mainly been considered
for military command and control systems, because a cen-
tralized network is often not feasible in such applications.
However, in recent years there is a growing interest in ad
hoc networks in other applications, such as peer-to-peer
computer communications, and communications between
mobile sensors (e.g. traffic safety systems).
As pointed out in [9], ad hoc networks pose many design
challenges. In this paper, we address the issue of resource
allocation when designing link access schemes. One ac-
cess scheme for ad hoc networks is Time Division Multiple
Access (TDMA), in which the transmission resource of a
radio frequency is divided into time slots, and a unit may
transmit in one or several time slots. It is known that,
although simple to implement, TDMA is very inefficient
from the resource utilization point of view. One possi-
bility to increase the network efficiency is to use Spatial
TDMA, or STDMA [16], which takes into account that ra-
dio units are usually spread out geographically, and hence
units with a sufficient spatial separation can use the same
time slot for transmission.
A number of specific planning problems, with differ-
ent levels of complexity, can be defined for access control
schemes in ad hoc networks. For example, one of the most
challenging problems is to design efficient, distributed al-
gorithms that can take traffic information into account,
and can handle fully mobile scenarios. Here, we will ad-
dress the problem of computing STDMA slot allocation
centrally for static networks without any knowledge of the
traffic distribution. This is one of the most basic planning
problems in this area. The objective of the optimization
problem is to minimize the length of the STDMA data
frame, such that all the units (or all the network links)
are assigned at least one time slot. Previous work for this
problem (e.g. [4], [6], [11], [14], and [17]) has focused on
heuristics. There is, therefore, a lack of methods that aim
at finding optimal solutions, which are important both
for evaluating heuristic algorithms, and for assessing the
true potential of STDMA. The main purpose of this pa-
per is to present set covering formulations to model re-
source optimization problems for both node-oriented and
link-oriented allocation strategies, and an efficient column
generation solution method. The proposed method gen-
erates optimal or near-optimal solutions in our numerical
experiments. The focus of this paper is not the method-
ology itself, but its application in context of ad hoc net-
works. Up to our knowledge, such an investigation has
not been conducted previously. We show that, in addi-
tion to theoretical analyzes of ad hoc network capacities
(such as [19]), mathematical programming methodologies
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
can aid as a powerful tool in rational ad hoc network de-
sign. Indeed, the results from this research can be used as
reference solutions when the complexity is increased, and
more realistic situations are tackled. Some of these future
research ideas are discussed in Section VIII.
The remainder of this paper is organized as follows. In
Section II we discuss two strategies of resource allocation.
The optimization problems are formalized in Section III,
and the computational complexity is studied in Section
IV. We present mathematical formulations in Section V,
and the column generation method in Section VI. Numer-
ical results are presented in Section VII, and conclusions
are drawn in Section VIII.
II. A
SSIGNMENT STRATEGIES
Resource allocation in STDMA can be node-oriented
(node assignment) or link-oriented (link assignment). In
the former strategy, a node is assigned one or several time
slots. In each of these time slots, the node may use any
of its links for transmitting data to another node. To
avoid access collisions, the nodes close to a transmitting
node are not allowed to transmit in the same time slot. In
addition, when a node is transmitting, it is required that
the accumulated interference on any of the outgoing links
does not exceed a certain threshold. It can be realized
that this assignment strategy is well-suited for broadcast
traffic.
In link assignment, a link is assigned one or several time
slots for communications between a specific pair of nodes
(point-to-point). Considerations for avoiding access col-
lisions and interference are similar to those for node as-
signment. Generally speaking, link assignment achieves
a higher spatial reuse of the time slots, but is, however,
less suitable for broadcast traffic. STDMA resource allo-
cation based on node assignment has been studied in [2],
[4], [5], and [6], while [3], [10], [14], and [17] focus on al-
gorithms using link assignment. Comparisons of the two
assignment strategies can be found in [11], [12], [13], and
[18].
III. P
ROBLEM DEFINITIONS
We will use the assumptions and notation in [10] when
defining the optimization problems. Consider an ad hoc
network consisting of a set of nodes N,whereeachnode
i ∈ N represents a radio unit. A direct communication
link can be established between nodes i and j if the cor-
responding signal-to-noise ratio (SNR) is greater than or
equal to a certain threshold, that is, if
SN R(i, j)=
P
i
L
b
(i, j)N
r
≥ γ
0
, (1)
where P
i
is the transmitting power of i, L
b
(i, j)isthe
path-loss between i and j,andN
r
is the effect of the
thermal noise. We use A to denote the set of (directed)
0 100 200 300 400 500 600 700 800 900
0
100
200
300
400
500
600
700
800
900
1000
x
y
Fig. 1. An ad hoc network with 20 nodes.
links. It has been commonly assumed (e.g., [12] and [13])
that P
i
= P,∀i ∈ N ,andL
b
(i, j)=L
b
(j, i), ∀(i, j) ∈ A.
We do not make these assumptions in this paper, because
our mathematical models and solution algorithms apply
also to the general case.
As the nodes are usually out-spread, an ad hoc network
is typically sparsely connected. A sample network of 20
nodes is shown in Figure 1.
There are several constraints and restrictions when as-
signing the STDMA time slots. First, a node can only
transmit or receive, but not both, in a time slot. Sec-
ondly, a node can only receive data from one other node
at a time, and finally, a link is error-free only if the signal-
to-interference ratio (SIR) is greater than or equal to a
threshold. For link (i, j), the last criterion can be ex-
pressed as
SIR(i, j)=
P
i
L
b
(i, j)(N
r
+
k∈N,k=i,j
P
k
L
b
(k,j)
)
≥ γ
1
, (2)
where the term
k∈N,k=i,j
P
k
L
b
(k,j)
is the accumulated in-
terference from other nodes.
The optimization problems for node assignment and
link assignment have the same input, namely a set of nodes
N, the path-loss between every pair of nodes L
b
(i, j), the
transmitting power P
i
of each node, the noise effect N
r
,
and the two threshold values γ
0
and γ
1
. The constraints
for the node assignment problem are the following.
• Two nodes that are connected by a link must be as-
signed different time slots.
• Two nodes, both having directed links to a third
node, must be assigned different time slots.
• A time slot can be assigned to a node only if all the
outgoing links of the node satisfy the SIR-constraints
(2).
The objective of the node assignment problem is to as-
sign at least one time slot to each node, such that the total
number of time slots is minimized.
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
For link assignment, the corresponding constraints are
the following.
• Two links that are adjacent, irrespective of the link
directions, must be assigned different time slots.
• A time slot can be assigned to a link only if the SIR-
constraint (2) for the link is satisfied.
The objective of the link assignment problem is to as-
sign at least one time slot to each link, such that the total
number of time slots is minimized.
IV. C
OMPUTATIONAL COMPLEXITY
We are able to show the following results for the opti-
mization problems defined in the previous section.
Proposition 1 The node assignment problem is NP-
hard.
Proof Consider the graph coloring problem defined for
an undirected graph G =(V, E). We construct, in poly-
nomial time, an instance of the node assignment prob-
lem, such that the two problems are equivalent. First,
for each edge (i, j) ∈ E, we introduce a node v
ij
.Let
V
E
= {v
ij
, ∀(i, j) ∈ E}. We then let N = V ∪ V
E
. The set
of directed links between the nodes in N , is defined as fol-
lowing. For a node v
ij
∈ V
E
, we define two directed links,
(i, v
ij
) and (j, v
ij
), and for every pair of nodes v
ij
∈ V
E
and v
kl
∈ V
E
, we defined a pair of directed links, (v
ij
,v
kl
)
and (v
kl
,v
ij
). We use A to denote the set of directed links.
We let γ
0
= γ
1
=1,P
i
=2, ∀i ∈ N ,andN
r
=1. For
each link (i, j) ∈ A,weletL
b
(i, j) = 1. For all the pairs of
nodes that do not have a link in A, we choose a sufficiently
large path-loss value, such that the SNR constraint for
these node pairs are not satisfied, and in addition, the
SIR constraints for links in A are redundant. (One path-
loss value that achieves this is 2|N |.) We then obtain an
instance for the node assignment problem, defined for the
node set N and link set A, along which the other problem
parameters defined as above.
To see the equivalence between the original node color-
ing problem and the derived node assignment problem, we
make the following observations. First, the nodes in set V
E
must be assigned different time slots (or colors). Secondly,
the node sets V and V
E
must use disjoint sets of colors.
Finally, two adjacent nodes in the graph coloring problem
cannot use the same time slot. These observations lead
directly to the conclusion that a feasible solution of one
problem corresponds to a feasible solution of the other
problem, and vice versa. For any of such pairs of solu-
tions, the difference in the objective values is a constant
(equals |E|). We further note that the reduction itself is
polynomial. Hence the conclusion.
Proposition 2 The link assignment problem is NP-hard.
Proof Consider the edge coloring problem defined for a
undirected graph G =(V,E) (known to be NP-hard). We
construct, in polynomial time, an instance of the link as-
signment problem, such that the two problems are equiv-
alent. For each edge (i, j) ∈ E, we define a direct link
from i to j. Denote the set of directed links by A.We
further let N = V . We then choose the values of trans-
mitting power, the noise effect, the path-loss parameters,
and the threshold values as in the previous proof. It is
easily realized that the derived link assignment problem
is equivalent to the edge coloring problem, and the con-
clusion follows immediately.
V. M
ATH E M AT I C A L FORMULATIONS
A. A Node-Slot Formulation
In this section, we present a node-slot formulation for
optimally assigning the time slots to the nodes. We use
T = {1, ..., |T |} to denote a set of time slots. To ensure
feasibility, it is sufficient to have |T | = |N|. We introduce
the following variables.
x
it
=
1 if time slot t is assigned to node i,
0 otherwise,
y
t
=
1 if time slot t is used,
0 otherwise.
The node-slot formulation (NSP) for STDMA node as-
signment is
[NSP] min
t∈T
y
t
(3)
t∈T
x
it
≥ 1, ∀i ∈ N, (4)
x
it
≤ y
t
, ∀i ∈ N, ∀t ∈ T, (5)
x
it
+
j:(j,i)∈A
x
jt
≤ 1, ∀i ∈ N,∀t ∈ T, (6)
P
i
/N
r
L
b
(i, j)
x
it
+ γ
1
(1 + M
ij
)(1 − x
it
) ≥ γ
1
(1 +
k∈N :k = i,j
P
k
/N
r
L
b
(k, j )
x
kt
), ∀(i, j) ∈ A, ∀t ∈ T, (7)
x
it
∈{0, 1}, ∀(i, j) ∈ A, ∀t ∈ T, (8)
y
t
∈{0, 1}, ∀t ∈ T. (9)
Here, the objective function (3) minimizes the total
number of assigned time slots. Constraints (4) ensure
that every node is assigned at least one time slot, and (5)
are the coupling constraints between the two sets of vari-
ables. Constraints (6) ensure that different time slots are
assigned to two nodes if they are connected by a link, or if
they both have links to a third node. The SIR-constraints
are defined in (7). Note that, if time slot t is not as-
signed to node i, i.e., x
it
= 0, then (7) is redundant for a
sufficiently large value of M
ij
. Otherwise, the constraint
becomes
P
i
/N
r
L
b
(i, j)
≥ γ
1
(1 +
k∈N :k = i,j
P
k
/N
r
L
b
(k, j )
x
kt
), (10)
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
which corresponds to (2).
The coefficient M
ij
canbesettoM
ij
=
k∈N :k = i,j
(P
k
/N
r
)/L
b
(k, j ), i.e., the sum of inter-
ference from all other nodes than i and j. However, not
all of the potential interfering nodes will be transmitting
at the same time, because the slot assignment of these
nodes must also satisfy constraints (6) and (7). Therefore,
it is possible to calculate a smaller value of M
ij
in order
to improve the LP-relaxation of NSP (we refer to [1] for
details).
One may also note that the formulation NSP contains
a lot of symmetry. For instance, there are many solutions
corresponding to the same assignment, but with differ-
ent time slots allocated. To break such symmetry, one
may add the constraints y
t
≤ y
t−1
,t =2, ..., |T |,which
imply that time slot t is assigned only if time slot t − 1
is assigned. Another type of symmetry is related to the
fact that swapping the nodes of any two time slots does
not affect the objective function value. One possibility to
handle this is to require that the slot assigned to node i
has an index less or equal to i, which can be stated as
x
it
=0, ∀i, t : i<t.
B. A Link-Slot Formulation
For link assignment, there is a link-slot formulation,
LSP, which is similar to NSP. In LSP, the cardinality of
the set of time slots can be set to the number of links (i.e.,
|T | = |A|). We use the following variables in LSP.
x
ijt
=
1 if time slot t is assigned to link (i, j),
0 otherwise.
y
t
=
1 if time slot t is used,
0 otherwise.
v
it
=
1ifnodei is transmitting in time slot t,
0 otherwise.
Formulation LSP is stated below.
[LSP] min
t∈T
y
t
(11)
t∈T
x
ijt
≥ 1, ∀(i, j) ∈ A, (12)
x
ijt
≤ y
t
, ∀(i, j) ∈ A, ∀t ∈ T, (13)
j:(i,j)∈A
x
ijt
+
j:(j,i)∈A
x
jit
≤ 1,
∀i ∈ N,∀t ∈ T, (14)
x
ijt
≤ v
it
, ∀(i, j) ∈ A, ∀t ∈ T, (15)
P
i
/N
r
L
b
(i, j)
x
ijt
+ γ
1
(1 + M
ij
)(1 − x
ijt
) ≥ γ
1
(1 +
k∈N :k = i,j
P
k
/N
r
L
b
(k, j )
v
kt
), ∀(i, j) ∈ A, ∀t ∈ T, (16)
x
ijt
∈{0, 1}, ∀(i, j) ∈ A, ∀t ∈ T, (17)
v
it
∈{0, 1}, ∀i ∈ N,∀t ∈ T, (18)
y
t
∈{0, 1}, ∀t ∈ T. (19)
Constraints (12) and (13) correspond to (4) and (5), re-
spectively. Constraints (14) state that two adjacent links
must be assigned different time slots. Variables x are cou-
pled to variables v using (15), and these two sets of vari-
ables are then used in the SIR-constraint (16). Similar
observations, as for NSP, concerning the coefficients M
ij
can be made for LSP. Moreover, for breaking symmetry,
we may add y
t
≤ y
t−1
,t =2, ..., |T |, and set x
ijt
=0if
the link index of (i, j) is less than t.
C. Set Covering Formulations
Formulations NSP and LSP are quite straightforward,
but not efficient from a computational point of view.
Even for a network of moderate size, the numbers of vari-
ables and constraints become very large. In our numeri-
cal experiments, the state-of-the-art mixed integer solver
CPLEX (version 7.0) is only able to solve the smallest in-
stance (a network with 20 nodes) of NSP to optimality.
For LSP, CPLEX did not manage to find a feasible integer
solution for any of our instances within reasonable time.
Instead of using formulations NSP and LSP, in this sec-
tion we reformulate the two optimization problems using
set covering formulations, for which a column generation
approach can be used for solving the linear programming
relaxations. This approach is similar to the one used in
[15] for solving graph coloring problems.
To obtain set covering formulations, we consider a group
of nodes (or links) that can transmit simultaneously. Such
a group defines a column in the set covering formulations.
To proceed, we let L
N
and L
A
denote the sets of all feasi-
ble groups of nodes and links, respectively, and define the
following variables related to the groups (or columns).
x
l
=
1 if column l is used,
0 otherwise.
For node assignment and link assignment, respectively,
let
s
il
=
1 if column l contains node i,
0 otherwise,
and
s
ijl
=
1 if column l contains link (i, j),
0 otherwise.
The set covering formulation for node assignment
(NSCP) is stated below.
[NSCP] min
l∈L
N
x
l
(20)
l∈L
N
s
il
x
l
≥ 1 , ∀i ∈ N, (21)
x
l
∈{0, 1} , ∀l ∈ L
N
. (22)
The objective is to minimize the number of columns
used, and the constraints (21) state that every node is
covered by at least one column. Below is the correspond-
ing formulation for link assignment.
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
[LSCP] min
l∈L
A
x
l
(23)
l∈L
A
s
ijl
x
l
≥ 1 , ∀(i, j) ∈ A, (24)
x
l
∈{0, 1} , ∀l ∈ L
A
. (25)
NSCP and LSCP have a very simple constraint struc-
ture. Note that the complexity of the problems lies mainly
in the construction of the sets L
N
and L
A
. For realistic
problem instances, the sizes of these sets are very large.
It is therefore not practical to generate all the columns in
the two sets a priori. To overcome this difficulty, we apply
a column generation solution approach.
VI. C
OLUMN GENERATION
NSCP and LSCP have, for realistic problem instances,
far too many columns to be handled directly by an integer
programming method, and the same holds for the corre-
sponding LP-relaxations. Column generation of the LP-
relaxation provides a decomposition of the problem into
master and subproblems. In this approach, columns are
left out because there are too many to handle efficiently,
and most of them will not be used in an optimal solution
anyway. To check optimality of the LP-relaxation, a sub-
problem, called the pricing problem, which is a separation
problem for the dual LP, is solved in order to identify new
columns to enter the basis. If such columns are found,
the LP is reoptimized. A classical application of the col-
umn generation technique is the
cutting stock problem
,first
presented in [7] and [8].
To solve the integer problem, branching is needed
when no columns price out to enter the basis and the
LP-solution does not satisfy the integrality conditions.
Branch-and-price allows column generation to be applied
throughout the branch-and-bound tree – and this must be
done – in order to guarantee an optimal integer solution.
For NSCP and LSCP, we can apply a branch-and-price
scheme, similar to that in [15], in order to ensure integer
optimality. However, in our computational experiments
we found that the LP-optimum was equal, or very close,
to the integer optimum. In addition, the columns needed
to solve the LP-relaxation were sufficient to enable opti-
mal or near-optimal integer solutions.
A. Node Assignment
The master problem for node assignment (NAMP), is
the LP-relaxation of NSCP with respect to a subset of
columns, L
0
N
⊆ L
N
. Initially, L
0
N
can be the set of
columns where each column contains one single node.
(This, in fact, corresponds to conventional TDMA.)
[NAMP] min
l∈L
0
N
x
l
(26)
l∈L
0
N
s
il
x
l
≥ 1, ∀i ∈ N, (27)
0 ≤ x
l
≤ 1, ∀l ∈ L
0
N
. (28)
When NAMP is solved, we need to identify new columns
to enter the basis, or possibly verify optimality, by ex-
amining whether any of the variables, x
l
,l ∈ L
N
\ L
0
N
,
has a negative reduced cost. This can be done by mini-
mizing the reduced cost ¯c
l
for all the columns in the set
L
N
\ L
0
N
. The reduced cost of column l can be expressed
as ¯c
l
=1−
i∈N
¯
β
i
s
il
, where
¯
β
i
, ∀i ∈ N, are the optimal
dual variables to (27). The minimum reduced cost can
then be expressed as
min
l∈L
N
\L
0
N
¯c
l
=1− max
l∈L
N
\L
0
N
i∈N
¯
β
i
s
il
. (29)
To solve (29), we formulate the column generation sub-
problem for node assignment (NASUB). We introduce the
following variables.
s
i
=
1ifnodei is included in the column,
0 otherwise.
NASUB can then be formulated as
[NASUB] max
i∈N
¯
β
i
s
i
(30)
s
i
+
j:(j,i)∈A
s
j
≤ 1, ∀i ∈ N, (31)
P
i
/N
r
L
b
(i, j)
s
i
+ γ
1
(1 + M
ij
)(1 − s
i
) ≥
γ
1
(1 +
k∈N :k = i,j
P
k
/N
r
L
b
(k, j )
s
k
), ∀(i, j) ∈ A, (32)
s
i
∈{0, 1}, ∀i ∈ N. (33)
If the optimal solution to NASUB results in a reduced
cost that is non-negative, the optimal LP-value to NSCP
is found. Otherwise, NAMP is reoptimized with a new
column added to L
0
N
, and the procedure continues. At
termination, the LP solution value is a lower bound to
the integer optimum of NSCP.
B. Link Assignment
Similar to NSCP, the LP-relaxation of LSCP can be
solved by column generation. The corresponding master
problem (LAMP), with a subset of columns L
0
A
⊆ L
A
,can
be formulated as follows.
[LAMP] min
l∈L
0
A
x
l
(34)
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
l∈L
0
A
s
ijl
x
l
≥ 1, ∀(i, j) ∈ A, (35)
0 ≤ x
l
≤ 1,l∈ L
0
A
. (36)
To find the minimum reduced cost among all neglected
columns, we want to find
min
l∈L
A
\L
0
A
¯c
l
=1− max
l∈L
A
\L
0
A
(i,j)∈A
¯
β
ij
s
ijl
, (37)
where
¯
β
ij
, ∀(i, j) ∈ A, are the optimal dual variables to
(35). This leads to the following column generation sub-
problem (LASUB),
[LASUB] max
(i,j)∈A
¯
β
ij
s
ij
(38)
j:(i,j)∈A
s
ij
+
j:(j,i)∈A
s
ji
≤ 1, ∀i ∈ N, (39)
s
ij
≤ v
i
, ∀(i, j) ∈ A, (40)
P
i
/N
r
L
b
(i, j)
s
ij
+ γ
1
(1 + M
ij
)(1 − s
ij
) ≥
γ
1
(1 +
k∈N :k = i,j
P
k
/N
r
L
b
(k, j )
v
k
), ∀(i, j) ∈ A, (41)
s
ij
∈{0, 1}, ∀(i, j) ∈ A, (42)
v
i
∈{0, 1}, ∀i ∈ N, (43)
where the variables have the following definitions.
s
ij
=
1 if link (i, j) is included in the column,
0 otherwise,
v
i
=
1ifnodei is transmitting,
0 otherwise.
As for the node assignment case, we obtain a lower
bound to LSCP when column generation terminates.
VII. N
UMERICAL RESULTS
We have used three networks obtained from the Swedish
Defense Research Agency. The networks contain 20, 40,
and 60 nodes, respectively. The numbers of links are 134,
184, and 396. The network with 20 nodes is shown in
Section III.
We use CPLEX (version 7.0) to solve the master
problems (NAMP and LAMP) in the column generation
method. In order to find integer feasible solutions, the
integer versions of NAMP and LAMP are solved to opti-
mality by CPLEX when column generation terminates.
A. Node Assignment
CPLEX was used to solve the node-slot formulation,
NSP, of these instances. For the network with 20 nodes,
CPLEX found the optimal solution with 16 time slots, in
a solution time of 1 second. For the other two networks,
however, CPLEX did not manage to find or to verify the
TABL E I
NUMERICAL RESULTS OF COLUMN GENERATION FOR NODE ASSIGNMENT.
Network LP IP Iterations Time Reuse
N20 16 16 10 3s 1.25
N40 14 15 43 32s 2.86
N60 26 26 60 271s 2.31
optimal solutions within the time limit, which was set to
10 hours. At termination, the best integer solutions use
15 and 29 time slots, respectively.
The results of the column generation method are sum-
marized in Table I, where LP shows the optimal value of
the LP-relaxation, and IP is the objective value of the
optimal integer solution of NAMP with respect to all the
columns needed for solving the LP-relaxation. The next
three columns in the table show the number of iterations,
the solution time, and the degree of spatial reuse, respec-
tively. The spatial reuse is defined as the ratio between
the number of time slots needed in conventional TDMA
and the number of time slots in the our integer solution.
For node assignment, the former equals |N|.
For the networks with 20 and 60 nodes, we are able
to find and verify the integer optimal solution. For the
network with 40 nodes, the integer solution and the LP-
bound differs by only one time slot.
B. Link Assignment
For the same network, the size of LSP is considerably
larger than that of NSP. For none of the LSP instances,
CPLEX was able to find a feasible integer solution within
reasonable time. A direct application of column genera-
tion also failed for the networks with 40 and 60 nodes,
because of the very long time (up to hours) needed for
solving the subproblem LASUB to optimality. We have
therefore made the following modification to the column
generation method. Instead of solving the subproblem to
optimality (which yields the column with the most neg-
ative reduced cost), we halt the branch-and-bound enu-
meration for the subproblem when two integer feasible
solutions have been found. If the reduced cost (of the
incumbent) is less than or equal to a threshold value of
-0.1, we terminate the solution process for the subprob-
lem, and add the incumbent column to the master prob-
lem LAMP. Otherwise, we resume the branch-and-bound
enumeration, which is halted again when another integer
solution is found, and so on. Solving the subproblem in
this fashion, we terminate the enumeration either when
a column with the reduced cost less than or equal to the
threshold value is found, or when it is verified that no
column with negative reduced cost exists.
With the above modification, the column generation
method was able to solve all the three instances. The
results are displayed in Table II.
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
TABL E II
NUMERICAL RESULTS OF COLUMN GENERATION FOR LINK ASSIGNMENT.
Network LP IP Iterations Time Reuse
N20 70 70 175 262s 1.91
N40 43 45 360 947s 4.09
N60 114 115 874 10936s 3.44
We observe that the lower bounds provided by the LP-
relaxation are very tight in all three cases. For one net-
work the integer optimum is found. For the other two
networks the integer solutions use one and two time slots
more than the LP-bounds, respectively.
VIII. C
ONCLUSIONS
We have addressed resource optimization in ad hoc net-
works using STDMA. In particular, we consider the prob-
lems of assigning time slots for node- and link-oriented as-
signment strategies. We show NP-hardness results for the
two problems, and present two linear integer formulations.
These two formulations are, as shown by our numerical ex-
periments, not efficient from the computational point of
view. To enable an efficient solution method, we develop
set covering formulations for the problems. Our numeri-
cal results show that the LP-relaxations of the set covering
formulations can be efficiently solved using a column gen-
eration approach, and that they provide very tight bounds
to the integer problems. In addition, column generation
enables integer solutions that are very close to the theo-
retical LP-bounds. That our approach can be used to gen-
erate valuable benchmarks for other heuristic algorithms
is, therefore, the main contribution of this work.
There are many remaining tasks and challenges for fu-
ture research. One issue to be addressed is a further val-
idation of the proposed method, by extending the com-
putational experiments for other network topologies and
instances. The next important step of our research is to
consider more complex problems, for example, resource
optimization that takes traffic distribution into account.
In particular, we note that the methodology developed in
this paper can be extended to provide benchmarks when
maximizing the network throughput. We also plan to in-
vestigate whether our method is helpful in studying capac-
ity regions of ad hoc networks [19]. We will then move into
two research directions. First, there is, from a practical
point of view, an interest to move from centralized algo-
rithms to distributed ones. Secondly, we need to explicitly
consider the time-varying properties of the medium, that
is, to tackle mobile scenarios in which the input data is dy-
namic due to the movements of the radio units. For these
complex problems, we will focus on simple but fast heuris-
tics which do not necessarily guarantee optimal solutions.
However, the optimal solutions for the basic scenarios,
such as those presented in this paper, serve as benchmarks
for future research in the area. Having access to these so-
lutions will certainly help in developing heuristic methods
for the complex problems.
A
CKNOWLEDGMENT
The authors wish to thank the research group at the
Department of Communication Systems, Swedish De-
fense Research Agency (FOI), for the technical discus-
sions and the test data. This work is partially financed by
CENIIT (Center for Industrial Information Technology),
Link¨oping Institute of Technology, Sweden.
R
EFERENCES
[1] P. Bj
¨
orklund, Two Resource Optimization Applications in Wireless
Networks, Licentiate thesis, Department of Science and Technology,
Link
¨
oping university, Sweden, 2002.
[2] I. Cidon and M. Sidi, “Distributed assignment algorithms for multi-hop
packet radio networks,” IEEE Transactions on Computers, vol. 38, pp
1353-1361, 1989.
[3] I. Chlamtac and S. S. Pinter, “Distributed nodes organization algorithm
for channel access in a multihop dynamic radio network,” IEEE Trans-
actions on Computers, vol. 36, pp 728-737, 1987.
[4] A.-M. Chou and V. O. K. Li, “Slot allocation strategies for TDMA pro-
tocols in multihop packet radio networks,” Proceedings of IEEE INFO-
COM ’92, pp 710-716, 1992.
[5] A. Ephremides and T. Truong, “Scheduling broadcasts in multihop radio
networks,” IEEE Transactions on Communications, vol. 38, pp 456-460,
1990.
[6] N. Funabiki and Y. Takefuji, “A parallel algorithm for broadcast schedul-
ing problems in packet radio networks,” IEEE Transactions on Commu-
nications, vol. 41, pp 828-831, 1993.
[7] P. C. Gilmore and R. E. Gomory, “A linear programming approach to the
cutting stock problem,” Operations Research, vol. 9, pp 849-859, 1961.
[8] P. C. Gilmore and R. E. Gomory, “A linear programming approach to the
cutting stock problem - part II,” Operations Research, vol. 11, 1963.
[9] A. J. Goldsmith and S. B. Wicker, “Design challenges for energy-
constrained ad hoc wireless networks,” IEEE Transactions on Wireless
Communications, pp 8-27, August 2002.
[10] J. Gr
¨
onkvist, “Traffic controlled spatial reuse TDMA in multi-hop radio
networks,” Proceedings of 9th IEEE International Symposium on Per-
sonal, Indoor and Mobile Radio Communications, pp 1203-1207, 1998.
[11] J. Gr
¨
onkvist, “Assignment methods for spatial reuse TDMA,” Proceed-
ings of IEEE Annual Workshop on Mobile and Ad Hoc Networking and
Computing (MobilHOC), pp 119-124, 2000.
[12] J. Gr
¨
onkvist, A. Hansson and J. Nilsson, “A comparison of access meth-
ods for multi-hop ad hoc radio networks,” IEEE Vehicular Technology
Conference Proceedings, pp 1435-1439, 2000.
[13] J. Gr
¨
onkvist, Assignment Strategies for Spatial Reuse TDMA, Licentiate
thesis, Royal Institute of Technology, Stockholm, Sweden, 2002.
[14] B. Hajek and G. Sasaki, “Link scheduling in polynomial time,” IEEE
Transactions on Information Theory, vol. 34, pp 910-917, 1988.
[15] A. Mehrotra and M. A. Trick, “A column generation approach for graph
coloring,” INFORMS Journal on Computing, vol. 8, pp 344-354, 1996.
[16] R. Nelson and L. Kleinrock, “Spatial-TDMA: A collision-free multihop
channel access control,” IEEE Transactions on Communications, vol. 33,
pp 934-944, 1985.
[17] C. G. Prohazka, “Decoupling link scheduling constraints in multihop
packet radio networks,” IEEE Transactions on Computers, vol. 38, pp
455-458, 1989.
[18] D. S. Stevens and M. H. Ammar, “Evaluation of slot allocation strategies
for TDMA protocols in packet radio networks,” Proceedings of the 1990
Military Communications Conference (MILCOM), pp 835-839, October
1990.
[19] S. Toumpis and A. J. Goldsmith, “Capacity regions for wireless ad
hoc networks,” IEEE International Conference on Communications,pp
3168-3173, 2002.
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
